Channel coding is an integral part of many communication systems. From the Shannon channel coding theorem, a fundamental performance limit of channels in such communication systems can be achieved by employing channel codes. Following Shannon's work in 1948, there have been significant developments in designing channel codes that can approach or achieve channel capacity. Such code design involves a mathematical description of a channel which, in the context of information theory, is equivalent to describing the channel as a set of transition probability density functions from an input channel alphabet to an output alphabet. Once a communication channel is represented with such an information theoretical model, code design techniques in coding theory can be employed to construct codes that can approach or achieve the capacity of the modeled channel. Numerous classes of codes have been developed to achieve the capacity of such channels. Among the most promising of these codes are Low Density Parity Check (LDPC) codes, Irregular Repeat-Accumulate (IRA) codes, convolutional codes, turbo codes, and polar codes.
Polar codes, for example, are proposed as channel codes for use in future wireless communications, and have been selected for uplink and downlink enhanced Mobile Broadband (eMBB) control channel coding for the new 5th Generation (5G) air interface, also known as the 5G New Radio (NR). These codes are competitive with state-of-the-art error correction codes and have low encoding complexity. See E. Arikan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051-3073, 2009. Successive Cancellation List (SCL) decoding and its extensions (e.g., SC List decoding) are effective and efficient options for decoding polar coded information.
Based on channel polarization, Arikan designed a channel code that is proven to reach channel capacity. Polarization refers to a coding property that, as code length increases to infinity, bit-channels (also referred to as sub-channels) polarize and their capacities approach either zero (completely noisy channel) or one (completely perfect channel). In other words, bits encoded in high capacity sub-channels will experience a channel with high Signal-to-Noise Ratio (SNR), and will have a relatively high reliability or a high likelihood of being correctly decoded, and bits encoded in low capacity sub-channels will experience a channel with low SNR, and will have low reliability or a low possibility to be correctly decoded. The fraction of perfect bit-channels is equal to the capacity of this channel.